sign in sign up
<<
Home , Hideki Yukawa

Part I Phenomenology of Elementary Particles 5 9/15/2009

2 Forces in The forces in nature are Gravity, the , the electromagnetic interaction (the latter two now unified as the electroweak interac- tion, the cornerstone of the ), and the strong or color interaction. All modern interaction models are based on the exchange of elementary quanta, in the simplest case tensor, vector, or scalar . Feynman graphs picture such exchange processes, but more than that, they associate precise terms for the amplitude describing the process with the various items in the graph: vertices (where lines meet) correspond to precisely prescribed vertex factors, internal lines to terms, and in- and outgoing particle lines to phase-space fac- tors dependent on particle type. μ 2.1 The Yukawa Interaction e In 1934 Hideki Yukawa proposed the existence of a light mass (bosonic field) that π would be responsible for (the mediator of) the short-range .2 The mass of the boson would be commensurate with the observed mean range of about 2 fm. The conjugate of the range would be the boson’s mass or energy: m=1/r = ħc/r = 197 MeV⋅fm / 2 fm ≈ 100 MeV. Powell et al. much later (1947) observed the Yukawa particle, the (140 MeV) or pi-, in photographic emulsions exposed to cosmic rays at high-altitude (mountaintops and balloons).3 Earlier, in 1937, the (105 MeV) had been discovered (at sea level),4 but was quickly recognized for what it was: a heavy version of the electron (a charged lepton), which did not have the , and thus could be hardly its messenger! In order to explain Yukawa’s approach, we turn to good old electromagnetism. Maxwell’s equations of electromagnetism can be succinctly described by the electromagnetic vector potential A and the scalar potential A0 (which simply equals the electric Coulomb potential V): JKJKJKJKdd ∇ ⋅=∇⋅=∇×+=∇×−=Eρ,0,0,, B EB BEj dt dt (I.4) JKd JK BAEA=∇×, =− −∇A0 dt The inhomogeneous Maxwell equations then become:

⎛⎞0 −−−ExyzEE ⎜⎟EBB0 − jjνμνμννμ≡=∂≡∂∂−∂=∂(,)0 j F() A A ⎜⎟xzy , (I.5) ⎜⎟EyzBB0 − x ⎜⎟ μμ μ ⎝⎠EBBzyx− 0 with definitions:

2 Hideki Yukawa, ”On the Interaction of Elementary Particles,” Proc. Physico-Mathematical Soc. of (3) 17, 48, pp 139-148 (1935); read 17 Nov 1934. 3 C.F. Powell et al., Nature 159, p186 and pp694-697 (1947). 4 S.H. Neddermeyer and C.D. Anderson, Phys. Rev. 50, 263 (1936); J.C. Streets and E.C. Stevenson, Phys. Rev. 52, 1003 (1937). μ

Part I Phenomenology of Elementary Particles 6 9/15/2009

JKJK ⎛⎞dddd⎛⎞ dμμν ⎛ d ⎞ ∂ ≡=∇∂=∂=−∇⎜⎟, , ,⎜⎟ , , and: g ⎜ , ⎟, (I.6) ⎝⎠dt dx dy dz⎝⎠ dt ⎝ dt ⎠ μ where we used the Bjorken & Drell5 choice for the metric gμν = Diag(1,–1,–1,–1); this contrasts, for instance, with Siegel’s choice6 gμν = Di- ag(–1,+1,+1,+1). For a stationary source j0=qρ(x), and only the scalar electric field potential is non-zero. Its equation of motion is: jj00=νμμμ= = qρ()r =∂ F 0 =∂( ∂ A 00 −∂ A) =−∇ 20 A , (I.7) =1,2,3 0 0∂ 0 because AAAμ ==A 0 andμμ ∂ = = 0 for a stationary source. ∂t The solution of this differential equation is: 1()qqρ r' Ad0(rr' )== , which, for a point charge ρδ ( rr )= ( ), becomes: A0 ( r ) (I.8) 44π ∫ rr'− π r This potential drops off like r-1, and is thus of “infinite” range. In order to obtain a short-range force, Yukawa added a mass term to the eq- uation, making the field A0(r) a “massive” one: 1()qρ r' −∇20AmAq + 20 =ρ(),rrr' with solution: A 0 () = d e−−m rr' , 4 ∫ rr'− (I.9) q which, for a point charge (rr )= ( ), becomes: Ae0 = −m r 4 r Thus, Yukawa created a short-range force, mediated by a massiveρδ carrier, with meanπ range r=|r|=1/m, due to the exponential fall-off. Be- cause the range of the strong force was known to be about 1-2 fm, Yukawa postulated the existence of a massive mediator of the strong force, and predicted it to have a mass around 1 to 1/2 fm-1, equal to ≈ħc/1 fm = 197 MeV. Another, more appropriate way of deriving this is to start with the relativistic mass-energy relationship, and “quantify” it by making the momentum and energy into operators: π 2 22 2 ∂∂⎛⎞ ∂22 22 Epm−=; using Ei → , and prr →− i , this becomes: ⎜⎟2 −+∇ mψψ ( , t ) =+() m ( , t ) = 0 (I.10) ∂∂ttr ⎝⎠ ∂ This equation is the famous Klein-Gordon equation, the relativistic version of the Schrödinger equation or the massive version of the Equa- tion of Motion for the electromagnetic field. In the stationary case, the time derivative vanishes, and, for the non-field-free case, the zero on the RHS is replaced by -αρ(r)ψ(r), where the –ve sign is chosen to make the potential attractive. For a point-like nucleon at rest, this equation has a time-independent solution

5 “Relativistic Quantum Mechanics,” J.D. Bjorken and S.D. Drell, McGraw-Hill, 1964. 6 “Fields,” by Warren Siegel, http://arxiv.org/abs/hep-th/9912205v3, Aug 2005. Part I Phenomenology of Elementary Particles 7 9/15/2009

α e−mr ψ ()r =− , with r=|r| and where α is a (+ve) proportionality “constant” (which could be a function of something other than r, and is 4π r in fact a function dependent on the nucleon spin as well), and the exponential has the mass m in it (in the proper units!). As expected, for m=0 the wavefunction is the Coulomb (photon) potential (field), which is massless. For m>0 the field/potential drops off exponentially lead- ing to the observed short range. The field has a mean range given by mr=1, or m≈100 MeV. A simple of the process is shown in the accompanying Figure. The amplitude described by the diagram is best expressed in the coupling constants (√α)2 and the “propagator” of the intermediate boson of mass m. The latter is expressed in momentum space rather than three-space with q≡ p2 – p1: 1 +∞12π π iqr2 iqr p1 p3 Ψ=Ψ−()qpp21 (πππ = ) dreddrdrrerψθφψ() = cos () Fourier ∫∫∫∫ Transform 4 space −100 (I.11) ∞∞∞e−mr 11 =====444r2() dr eiqr dr re−− r m iq dr re − rs ∫∫∫LaPlace 222 q ≡ p3−p1 000rsmqTransform + The amplitude for the process is thus proportional to α/(m2 + q2) and the cross section to the square of this. x The exact same effect is responsible, we now know, for the short range of the weak force: the very massive t W (80,403±29 MeV) and Z (91,187.6±2.1 MeV) bosons have an extremely short range, and thus also limit p2 p4 (weaken) the interaction. The fact that the mass of many weakly decaying particles is much smaller than the 2 2 –5 W mass causes the q to be very small compared to mW , and therefore the weak decay amplitude appears to be constant and small: G ≈ 10 –2 2 –1 GeV ≈ αW/mW and thus mW was expected to be around 30 GeV for αW ≈ α = (137) . Initially the pion was thought to explain the character of the nucleon-nucleon force, the strong interaction, completely. However, when more experimental data became available, it turned out that the experimentally observed shape and behavior of the interaction was in disagree- ment with the predictions: at the very small distance significant discrepancies exist, and the repulsive part of the strong force cannot be ex- plained with pion exchange model. Only at the larger distances, r>2 fm or so, is good agreement seen. Attempts were made to mix in other (by then observed) heavier (the ρ, ω, ϕ, f, g and many others), in other to modify the short-range behavior, with good but not full success. With the advent of even higher energy accelerators, substructure was discovered in the nucleon, and also in the mesons: the quarks. It is as- sumed that the full knowledge of the quark potential in the nucleon would allow us, in principle, to explain the form of the nucleon-nucleon interaction as a kind of spillover “van der Waals” color-dipole interaction of the basic quark- interactions going on inside the nucleon. However, no one has succeeded yet in calculating the nucleon-nucleon potential purely based on the quark model: calculations are impossi- bly complex so far. A promising avenue is the method of “Lattice Gauge” calculations, which requires high-speed supercomputers for great amounts of calculations. Part I Phenomenology of Elementary Particles 8 9/15/2009

2.2 Gravity On the scale of all other forces, gravity is extremely weak at the scale. This presents a huge problem for theories that at- tempt to unify all forces together. In such a scenario, the forces at the unification energy scale should become all equal, and judging from the extrapolation of the strength of the forces, strong and electroweak interactions unify at energies of about 1-100 TeV (if supersymmetry is 19 correct!), whereas gravity joins them only at the Planck scale, MP≈10 GeV, which is way beyond accessibility. In the same way as the di- 2 2 mensionless coupling for electromagnetism equals α = e /(4πε0)/ ħc = 1/137, we have the gravitational coupling αG = Gm / ħc: Gm2 = c 197 MeV⋅× fm 3.16 10−26 Nm 2 αα=≈===; for 1, we find m2 kg2 , GG=cG6.67×⋅ 10−−11 Nm 2 kg 2 6.67 × 10 − 11 Nm 2 (I.12) hence: m ≈× 2.2 10−819 kg =× 1.2 10 GeV Thus, the Planck scale is the energy at which gravitational energy becomes comparable to the mass energy: m2 1111 Gm==, thus for r one obtains: m =≈ = (I.13) rm G 6.67×× 10−−−11 Nm 2 /kg 2 6.71 10 39 GeV 2 This huge difference in scales is quite mystifying, and problematic for unification. Very recently, a new development – still regarded very preliminary – has come about, which is based on more, but infinitesimally small, dimensions, which are accessible to gravity (as a space- time geometrical force) but not to the other interactions. If some of these extra dimensions are not extremely small, then unification might occur at the TeV scale, and at the same time, spectacular effects are expected at the LHC and possibly the Tevatron. Sub-millimeter gravity experiments are underway, and currently reach 140 µm without detecting deviations from the 1/r2 law.7

7 E. Adelberger et al., see http://mist.npl.washington.edu/eotwash/ Part I Phenomenology of Elementary Particles 9 9/15/2009

3 Symmetries and Conservation Laws A mechanical system can be described by a Lagrangian that fully defines the system and its interactions. The equations of motion of the sys- tem (and its various parts), can be derived from Hamilton’s principle, which yields the Euler-Lagrange as the equations of motion. The La- grangian formalism is also applicable in quantum mechanics, and is the formalism of choice for particle theorists. In the Lagrange formal- ism, conserved quantities (Energy, momentum, angular momentum) are a direct result of certain invariances of the Lagrangian under trans- formations of space and time variables. For example, if the Lagrangian is invariant under a shift in the time variable (homogeneity of time): L(t) = L(t’) for t’ = t + δt, then it follows that the total energy of the system is a conserved quantity. Similarly, invariance under translations (homogeneity of space) leads to momentum conservation, and invariance under rotations (isotropy of space) leads to conservation of angular momentum. Other “symmetries” exist for the Lagrangian that describes the interactions of elementary particles: for instance, the electromagnetic interac- tion is invariant under the inversion of all the coordinates of the system (r → r’ = −r). This is an example of a discrete symmetry. The cor- responding conserved quantity is the parity (P) of the system, which can have +1(even) or –1(odd) eigenvalues: P(ψ(r)) = ψ(−r) = ±ψ(r). Both the electromagnetic and strong interactions conserve the parity of a system, whereas the weak interaction does not! In the Standard Model exist a few conserved quantities that do not correspond to any known symmetry or invariance of the Standard Model Lagrangian. These are the conservation of number B and lepton numbers Le, Lμ, and Lτ, which formalize the empirical observation that the sum of quarks and antiquarks, and the sum of leptons and antileptons is conserved in interactions. Defining the quark to have a ba- 1 ryon number of /3, and the electron and the electron-neutrino to have an electron-number of 1 (while their antiparticles have the opposite quantum numbers!), baryon and lepton number conservation “explain” the occurrence of processes like: − n → p + e +⎯νe − − μ → νμ + e +⎯νe − − π → μ + ⎯νμ and “forbid” many non-observed processes like: n → p + e− − + n → π + π − − μ → e +⎯νe π− → μ− + e− + e+

Symmetries and the conservation laws that follow from them are important tools to constrain the possible forms that the Hamiltonian (or Lagrangian) of a system can take. For example, the requirement of Lorentz invariance of the Lagrangian describing a system narrows down the possible options for the Lagrangian to a limited number of scalar combinations of Lorentz (pseudo-) scalars, (pseudo-) vectors and ten- sors. Thus, the empirical observation of conservation laws means that the correlated symmetry law must be obeyed by the Hamiltonian (or Lagrangian) describing the system, and a lot can be learned about the possible forms the Hamiltonian is allowed to have. From the derived symmetries, one can then extrapolate and try to find the underlying guiding principles of the theory describing the system. Part I Phenomenology of Elementary Particles 10 9/15/2009

In the following we will use the non-relativistic Schrödinger equation for illustrating the procedures for finding conserved quantities and their associated symmetries (invariances under transformations of the wave functions in the Schrödinger equation). However, the very same reasoning applies in the fully relativistic description, which typically uses the Lagrangian formalism, and is decidedly more cumbersome. In the Schrödinger picture, the observables are time-independent operators, while the wave functions depend in general on the time parame- ter (the reverse is true for the Heisenberg picture).

3.1 Conserved Quantities The Schrödinger equation with Hamiltonian H and wave function ψ(r,t) is: ∂ itψψ(,)rHr= (,) t (I.14) ∂t Observables are Hermitian operators that act on the wave functions describing the system. Because an observable F (time-independent in the Schrödinger formalism) leads to a measurement value, the expectation value 〈F〉 must be real: * * * F≡==∫∫∫∫ dVψψ***†*††FFFFFF ψ ψ Fψψ ψ dV( ψ) ()ψψ = ψ dV ψ( ) = dV , thus: = (Hermitian!) (I.15) In the general case, the expectation value 〈F〉 changes with time. Consider now the condition under which this expectation value 〈F〉 is con- stant, i.e. a conserved quantity: dd dψ * dψ FdVdVdV==+**FFF (I.16) dt dt∫∫ dt ∫ dt Using the Schrödinger equation and its complex conjugate equation: * ∂∂ψψ* ii=−==HHH and ()* (I.17) ∂∂ttψψψ because H, being an observable, is Hermitian. Filling in the derivatives of the wave function: d FdVi=+−=−=()ψ **HFψψ dVi F() H ψ idV ψ ψ * HFFH (ψ ψ idV ) *[] HF,, (I.18) dt ∫∫ ∫ ∫ with [H,F]≡HF-FH the commutator of H and F. Thus the expectation value of F is conserved if and only if the commutator [H,F]=0. Be- cause an operator commutes with itself, the Kinetic energy operator and the “speed operator” commute: I can measure the kinetic energy and the speed simultaneously. A particle in a box, which has only elastic collisions with the wall, will conserve energy, and the speed is a conserved quantity. This is not so for the position measurement: the measurement of “x” is not commuting with the measurement of energy (the energy operator is p2/(2m)= –∇2/2m, which clearly not commutes with x).

3.2 Symmetry Transformations A symmetry transformation is a transformation of the wave function that leads to a new wave function that is again a solution to the same Schrödinger equation as the original wave function. Consider a simple transformation U: Part I Phenomenology of Elementary Particles 11 9/15/2009

ψ '(rUr ,tt )= ψ ( , ) (I.19) This transformation must leave the norm of the wave functions ψ and ψ’ the same: * ∫∫dVψψ** ψ ψ ψ == ψ ψ dV' ' ∫ dV()()U U = ∫ dV *†† UU , thus UU == 1 or U †1 U− (I.20) i.e. U is a unitary operator: any transformation is represented by a unitary operator. Now consider a symmetry transformation; then, the wave function Uψ satisfies the same Schrödinger equation as ψ: ∂()Uψ ∂ ii====−==H() U, and thus: U−−−111 HU H , or: U HU H , and: UU HU UH 0[] H , U (I.21) ∂∂tt Thus, if U is also ψψψHermitian it will be an observable with a conserved expectation value! However, only some unitary transformation opera- tors are also Hermitian. Examplesψ of such are discrete (non-continuous) symmetry transformations such as the Parity transform: Pψ(r,t)= cψ(–r,t), or the charge conjugation operator which changes particle into anti-particle: C|e-〉=c|e+〉. Because these operators turn the wave function back into itself after a second transformation, they necessarily have U2=1. Therefore, they are both unitary and Hermitian because: UU=1=U–1U=U†U or U†=U.

3.2.1 Constructing a Conserved Quantity from a Symmetry Operation In general H is poorly known, and commuting operators F cannot be easily determined by construction. However, we can establish that [H,F]=0 if we can find a symmetry transformation that leaves H invariant. In the preceding section, we saw that some discrete symmetry transformations satisfy this criterion: they are both a symmetry operator and an observable of the system, and their expectation value is a conserved quantity. Continuous transformations U form a group that includes the unity transformation 1. The elements of this group can be arbitrarily close to the unity transformation. A general continuous unitaryκ transformation may be written as a succession of transformations infinitesimally close to 1: n κ 23 ⎛⎞ ()iiκκFF() iκF UFF=+lim⎜⎟ 1ii =+++ 1 += ... e. (I.22) n→∞ ⎝⎠n 2! 3! This expansion shows clearly that the transformation U will be unitary if F is Hermitian: †† UU†()†= eeiiκκFF−−== κ e i FF 1 if and only if F = F . (I.23) In addition, if U commutes with H, so will F. This is easiest shown for small parameter ε, i.e. transformations infinitesimally close to the identity transformation. Then, the series expansion in equation (I.22) can be broken off at the second term, and: 0,=[HU] =−=+ HU UH H1 (iiiεεε F )() −+ 1 FH =[ HF ,] ⇒[ HF ,0] =, (I.24) so that the observable (Hermitian operator) F has a conserved expectation value.

Example 1: Conservation of Linear Momentum Consider a wave packet ψ(x). Suppose the existence of a transformation operation U(Δx), which transforms the wave function into ψ’(x): Part I Phenomenology of Elementary Particles 12 9/15/2009

ψ '(x )= U (Δxx )ψ ( ) , (I.25) see the adjacent Figure. In the case of ψ’(x), the operation is a pure translation, but in general the transformation can do any- ψ ’(x) thing, e.g. produce ψ’’(x)! ψ ’’(x) However, if U(Δx) is a symmetry transformation, both ψ(x) and ψ(x) ψ’(x) satisfy the same Schrödinger equation, and [H,U]=0; inva- riance means that ψ does not change shape under U: ∂∂ψ '(x ) ⎛⎞ ψ ()x =+Δ≈+Δ=+Δψψ '(xx ) '() x ψ x⎜⎟ 1 x '() x (I.26) x ∂∂xx⎝⎠ Δx where we used the Taylor expansion. Multiplying from the left x x +Δx by (1–Δx(∂/∂x)), and ignoring terms of order Δx2 and higher 0 0 gives: 2 ⎛⎞⎛⎞⎛⎞∂∂∂∂⎛⎞2 ⎜⎟⎜⎟⎜⎟1()11'()1'()'()−Δx ψψψψxx ≈ −Δ +Δ xxx =⎜⎟ −Δ2 xx ≈ . (I.27) ⎝⎠⎝⎠⎝⎠∂∂∂∂xxxx⎝⎠ Comparing Equations (I.27) and (I.25) gives the expression for U(Δx): ∂∂i −Δxixi Δ −Δxp ⎛⎞∂ ∂∂xx= x ∂ Up()Δ=xxeee⎜⎟ 1 −Δ ≈ = = , with: x ≡− i= ; (I.28) ⎝⎠∂x ∂x i.e.: the corresponding observable is the momentum operator (in the x-direction), which has a conserved expectation value. Note that the units of the exponent vanish as they should. (Note, the full Taylor series expansion ψ(x)=exp[Δx(∂/∂x)]ψ’(x) could have been used as well.)

Example 2: Invariance under Rotations Consider an infinitesimal rotation through an angle δθ in 3-d space: ∂ JK R()δψθ ()rr ψ=+⋅ δ ψ () ψ δθ ψ ()rr =+⋅×∇ ()θ ()rrr () (Taylor expansion) ∂θ (I.29) JKi JK JK Rii()θθ=1 + ⋅()r ×∇ = 1 +θ ⋅LLr , with ≡−==() ×∇ =− ∇ δδ δrr= θ n ⎛⎞i θ i Thus R()θθ=+⋅= lim1⎜⎟LL exp(z ), for example, for a rotation of angular size θ around the z-axis, and again [H,Lz]=0 if the Ha- n→∞ ⎝⎠= n = miltonian is invariant for rotations around the z-axis. As application, consider the production of a ρ0-meson in π+π− collisions: total angular momentum J is conserved in this strong interaction. Choosing the z-axis along the π+π−-directions in the center-of-mass system, with the fact that the pion spins are zero, one concludes that the + − 0 component of L along z-axis must vanish: Lz=0. The angular distribution of π π -pairs from the subsequent decay of the ρ -meson, I(θ,φ),